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Monday, September 19, 2011

Everything is a Parabola

Lined up the kids at the board along a big number line. Everybody picked a number : their "x." I said, "Shawn's x is at 2. However far away you are from Shawn, move that many floor tiles into the room, perpendicular to the board." They move into a lovely approximation of the graph of f(x) = |x - 2|. "Hey what kind of a shape are you guys making?" "A PARABOLA!" Practically in unison. FACEPALM.

12 comments:

KateGrab this and run with it. This can provide some lovely conversations about the similarities between the graphs/functions. You can talk about the abs(x) = sqrt(x^2) here. You can talk about the vertex form and its similarities here. The facepalm (I always flash back to Woody Allen in Annie Hall) is the perfect first reaction. Then the second is to run with the mathematics here.

Because Algebra 1 is force fed in earlier courses, and parabolas are defined loosely as "that U shape".

I've been feeling very down on Algebra as a course lately. I wish we could give it the treatment that Geometry and Calculus (ideally) get: a constructed course, highlighting the subject as a system to be explored, using the rules of logic.

@mrdardy, I did that a little bit, "you want to see a parabola? i'll show you a parabola!" I had them move the square of their distance from the anchor kid. But then I moved on with my graphing absolute value lesson. For me, "run with it" turns into "me talking" and I don't like "me talking" lessons. I'm the only one who gets much out of it.

@Scott, Yes. I got to see what that would look like a little bit at PCMI this summer. I also began to understand the wisdom of integrated curricula. I am on the lookout for a text that treats Algebra that way. I ordered a copy of the CME project books and have high hopes.

@Shawn and Hank, I hear you. Even linear patterns get called exponential, because they increase, I guess? I think they just enjoy knowing and remembering big fancy words. I do the same thing - it makes one feel smart.

Kate I know what you mean there. The way I try to combat this - and I know it is not always successful - is that if I am the one doing most of the talking, I try to make sure that most of what comes out of my mouth is in the form of a question or a reminder of what they have seen before.

Before these students got to your class, the only graphs they saw that change direction are parabolas. It's pretty natural for them to think so, then. The remedy, to me, is to present more general graphs as early as possible. (I am an author on the CME curriculum, and even before students learn equations of lines, they graph y = x^2 and y = |x| and x^2 + y^2 = 25, among others. A graph is just the set of points that make the corresponding equation true.)

Later on, you can take it in a few other directions, such as looking at the graph of y = x^{2/3}. It has the same but different behavior. I also like looking at non-functions that use absolute value notation, like |x| + |y| = 5 compared to x^2 + y^2 = 25.

Over generalization is always a standard part of learning; teachers should never be surprised when it happens. It is especially common in math, because good teachers tell kids to look for patterns – so they do. The key is how we react. Do we show them the subtle differences (which it sounds like you did) or do we get aggravated and try to drill the difference into them?

I think what makes this feel so frustrating is that we expect students to make connections and look for patterns, and they do. *But* only to stuff they have seen in another math class. They have a wealth of background knowledge and real-world experience (including shape recognition) that they could tap into, but somehow they have gotten the idea that math is disconnected from all of that, and pick the closest thing that they've seen in a previous math class.